- The paper shows that Gemma 3 4B IT verbally conflates contingent falsehood with logical impossibility, achieving 55.3% accuracy on four-way philosophical classification and labeling 12 of 15 contingent falsehoods as contradictions.
- The paper finds a double dissociation in residual-stream activations: truth decoding reaches 0.93 balanced accuracy, while impossibility decoding reaches 0.97 and separates impossible from false statements with 1.00 balanced accuracy.
- The paper shows that necessary falsehoods cluster closer to semantic anomalies than ordinary falsehoods, while emphasizing that these correlational probes do not establish causal use, generality beyond one model, or applicability across languages and tasks.
Overview and motivation
This paper asks whether a LLM internally distinguishes statements that are false from statements that could not be the case under any admissible interpretation. The distinction is classical: Aristotle's principle of non-contradiction, Wittgenstein's account of sense in the Tractatus, and Chomsky's demonstration that grammaticality can survive semantic breakdown all mark different ways in which language can fail to describe anything. Prior interpretability work established that the truth value of ordinary factual statements is linearly decodable from transformer residual streams (Cunningham et al., 2023, Lieberum et al., 2024), but no prior study had asked whether impossibility is represented as anything other than an extreme point on that truth axis. A model could, in principle, treat "a married bachelor lives here" as merely a very confident instance of "Paris is the capital of Germany."
The author tests this with Gemma 3 4B IT (Team et al., 25 Mar 2025) using two stimulus sets: 85 prompts drawn from 17 philosophical families (contradictions, definitional impossibilities, liar-style paradoxes, sorites, identity puzzles, semantic anomaly), and a topic-matched modality set of 15 topics each realized in five conditions — true, contingently false, improbable but possible, semantically anomalous, and necessarily false. All five conditions of a family share topic vocabulary, and necessary falsehoods deliberately vary in construction (reflexive comparison, temporal reversal, part–whole counting, definitional violation, kinship circularity, explicit contradiction), so no single lexical template identifies the class.
Behavioral conflation of falsehood and contradiction
The model's verbal classifications collapse empirical falsehood into logical contradiction. On the philosophical set it achieved only 55.3% exact four-way accuracy, defaulting heavily to "paradox" as a broad rejection label (58 of 85 prompts). On the modality set, it labeled 12 of 15 contingent falsehoods "contradiction," placing "Paris is the capital of Germany" and "whales are fish" in the same verbal category as married bachelors. Its one-sentence explanations used the same idiom for both, e.g., describing a false statement about apples as contradicting "established biological knowledge." Improbable claims were also frequently rejected (6 of 15 labeled contradiction), indicating over-broad rejection rather than fine-grained modal judgment.
A double dissociation in activation space
The internal representation tells a different story. Three L2-regularized logistic probes were trained at every depth with grouped cross-validation holding out entire topic families: an impossibility probe, a truth probe, and an anomaly probe. The truth probe peaked at 0.93 balanced accuracy, consistent with earlier truth-probing results [2310.xxxxx]. The impossibility probe peaked at depth 16 (transformer layer 15) with balanced accuracy 0.97 (family-restricted permutation P=0.0005; Bonferroni-adjusted over 35 depths, P=0.018).
The decisive result is transfer across contrasts, which forms a double dissociation:
| Probe direction |
False vs. true |
Impossible vs. true |
Impossible vs. false |
| Truth |
0.96 |
0.93 |
0.20 |
| Impossibility |
0.51 |
0.98 |
1.00 |
| Anomaly |
0.49 |
0.98 |
0.96 |
Along the direction separating true from false, necessary and contingent falsehoods are indistinguishable; along the impossibility direction they are separated nearly perfectly, against a strongest surface baseline of 0.77 (character TF–IDF). The two probe directions are close to orthogonal throughout the network, with absolute cosine at most 0.12 beyond depth 10. Whatever the model tracks when distinguishing the impossible from the false is not what it tracks when distinguishing the false from the true.
Cross-dataset transfer was partial: the impossibility probe trained on the modality set separated coherent controls from targets in the heterogeneous philosophical set at AUC up to 0.72, and reverse transfer reached 0.79. This indicates shared structure but not a single unified signal spanning plain necessary falsehoods and self-referential paradoxes.
Proximity to semantic anomaly
The anomaly direction, trained only on Chomsky-style selectional violations, separates impossible from false statements at AUC 0.96, and has a cosine of roughly 0.4 with the impossibility direction in middle layers while remaining distinguishable from it (impossible vs. anomalous separable at AUC up to 0.89). In this activation space, necessary falsehoods sit nearer the operational category of semantic anomaly than to ordinary falsehood — the model groups "taller than itself" with "colorless green ideas" rather than with "capital of Germany."
Sparse autoencoder features at layer 15 repeat this geometry. In the higher-capacity Gemma Scope 2 checkpoint (~90 active features per prompt), the features most selective for impossibility fire predominantly on anomalous sentences (prevalence 0.40–0.67) and impossible ones (0.67–1.00) but rarely on contingent falsehoods (0.07–0.80, with the most selective feature at 0.07). No individual feature in the sparser checkpoint (~16 active features) separated impossible from false, suggesting the signal is real but distributed rather than localized in a single feature at this dictionary size. These SAE candidates are correlational, derived from the same 160 prompts, with no independent corpus or inferential statistics attached to individual features.
Limitations and open questions
The paper is explicit about its constraints. Linear decodability establishes accessibility, not causal use (Sahoo et al., 1 Jun 2026); nothing shows the model consults these directions when answering. The stimuli are English-only, use a single template, and number only 160; the fifteen topic families cannot exhaust the space of impossibility constructions. Surface baselines of 0.67–0.77 show that lexical form explains part of the separability, though visibly not all of it. The "necessary falsehood" label is operational — some items depend on definitions or background constraints rather than pure formal contradiction — and the reported geometry concerns representational similarity, not the metaphysics of impossibility or Wittgenstein's taxonomy of sinnlos versus unsinnig. The behavioral/internal dissociation is documented in one small model; whether larger models sharpen or dissolve the distinction remains open, as does whether the observed directions play any functional role in generation.
Conclusion
This study documents a concrete dissociation between what a model says and how it represents: Gemma 3 4B IT verbally conflates contingent falsehood with contradiction, yet its residual stream carries truth and impossibility along nearly orthogonal directions, with necessary falsehoods represented closer to semantic anomaly than to ordinary falsehood. The result is a modest, correlational empirical footnote to a long-standing philosophical distinction — sufficient to show that two human categories come apart in one activation space, but not to map the model's own organization of the sayable and the unsayable.
